Abstract
Let A be a finite-dimensional algebra over an algebraically closed field \(\Bbbk \). For any finite-dimensional A-module M we give a general formula that computes the indecomposable decomposition of M without decomposing it, for which we use the knowledge of AR-quivers that are already computed in many cases. The proof of the formula here is much simpler than that in a prior literature by Dowbor and Mróz. As an example we apply this formula to the Kronecker algebra A and give an explicit formula to compute the indecomposable decomposition of M, which enables us to make a computer program.
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Acknowledgements
The starting idea was made when the first named author was at the lecture on the Kronecker canonical form of singular mixed matrix pencils by S. Iwata at CREST meeting in February, 2016. In fact, later we recognized that Iwata–Shimizu [8] had given relationships between values of ranks \(p_n(M)\) (resp. \(i_n(M), r_n(0, M)\), and \(r_n(\infty , M)\)) in Definition 2 and the values of \(\varvec{d}_M(L)\) for \(L = P_n\) (resp. \(I_n, R_n(0)\), and \(R_n(\infty )\)) in Theorem 2. However, they did not mention regular indecomposables \(R_n(\lambda )\) with \(\lambda \ne 0, \infty \). The authors would like to thank S. Iwata for his lecture and M. Takamatsu for informing us of the paper [8]. After submitting the paper E. Escolar and the referee pointed out that the paper [6] is already published dealing with the same problem with similar results, for which we are very thankful. This work is partially supported by Grant-in-Aid for Scientific Research 25610003 and 25287001 from JSPS (Japan Society for the Promotion of Science), and by JST (Japan Science and Technology Agency) CREST Mathematics (15656429).
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Asashiba, H., Nakashima, K. & Yoshiwaki, M. Decomposition theory of modules: the case of Kronecker algebra. Japan J. Indust. Appl. Math. 34, 489–507 (2017). https://doi.org/10.1007/s13160-017-0247-y
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DOI: https://doi.org/10.1007/s13160-017-0247-y